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Annuities are structured payment plans that an individual receives from a different party. This other party, generally, is the company that acts on another person’s behest, such as a financial agent or a government agency. The pertinence is to the payment of a particular sum to the individual or a period of several years instead of a single lump su
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Simply put, time value of money is the value of money figuring in a given amount of interest for a given amount of time. For example 100 dollars of today's money held for a year at 5 percent interest is worth 105 dollars, therefore 100 dollars paid now or 105 dollars paid exactly one year from now is the same amount of payment of money with that given interest at that given amount of time. This notion dates at least to Martín de Azpilcueta of the School of Salamanca.
The method also allows the valuation of a likely stream of income in the future, in such a way that the annual incomes are discounted and then added together, thus providing a lump-sum "present value" of the entire income stream.
All of the standard calculations for time value money derive from the most basic algebraic expression for the present value of a future sum, "discounted" to the present by an amount equal to the time value of money. For example, a sum of FV to be received in one year is discounted (at the rate of interest r ) to give a sum of PV at present: PV = FV — r · PV = FV/(1+r).
Some standard calculations based on the time value of money are:
Calculations
There are several basic equations that represent the equalities listed above. The solutions may be found using (in most cases) the formulas, a financial calculator or a spreadsheet. The formulas are programmed into most financial calculators and several spreadsheet functions (such as PV, FV, RATE, NPER, and PMT).
For any of the equations below, the formula may also be rearranged to determine one of the other unknowns. In the case of the standard annuity formula, however, there is no closed-form algebraic solution for the interest rate (although financial calculators and spreadsheet programs can readily determine solutions through rapid trial and error algorithms).
These equations are frequently combined for particular uses. For example, bonds can be readily priced using these equations. A typical coupon bond is composed of two types of payments: a stream of coupon payments similar to an annuity, and a lump-sum return of capital at the end of the bond's maturity - that is, a future payment. The two formulas can be combined to determine the present value of the bond.
An important note is that the interest rate i is the interest rate for the relevant period. For an annuity that makes one payment per year, i will be the annual interest rate. For an income or payment stream with a different payment schedule, the interest rate must be converted into the relevant periodic interest rate. For example, a monthly rate for a mortgage with monthly payments requires that the interest rate be divided by 12 (see the example below). See compound interest for details on converting between different periodic interest rates.
The rate of return in the calculations can be either the variable solved for, or a predefined variable that measures a discount rate, interest, inflation, rate of return, cost of equity, cost of debt or any number of other analogous concepts. The choice of the appropriate rate is critical to the exercise, and the use of an incorrect discount rate will make the results meaningless.
For calculations involving annuities, you must decide whether the payments are made at the end of each period (known as an ordinary annuity), or at the beginning of each period (known as an annuity due). If you are using a financial calculator or a spreadsheet, you can usually set it for either calculation. The following formulas are for an ordinary annuity. If you want the answer for the Present Value of an annuity due simply multiply the PV of an ordinary annuity by (1 + i ).
Formula
Present value of a future sum
The present value formula is the core formula for the time value of money; each of the other formulae is derived from this formula. For example, the annuity formula is the sum of a series of present value calculations.
The present value (PV) formula has four variables, each of which can be solved for:
- PV is the value at time=0
- FV is the value at time=n
- i is the rate at which the amount will be compounded each period
- n is the number of periods (not necessarily an integer)
The cumulative present value of future cash flows can be calculated by summing the contributions of F V t , the value of cash flow at time=t
Note that this series can be summed for a given value of n, or when n is
. This is a very general formula, which leads to several important special cases given below.
Present value of an annuity for n payment periods
In this case the cash flow values remain the same throughout the n periods. The present value of an annuity (PVA) formula has four variables, each of which can be solved for:
- PV(A) is the value of the annuity at time=0
- A is the value of the individual payments in each compounding period
- i equals the interest rate that would be compounded for each period of time
- n is the number of payment periods.
To get the PV of an annuity due, multiply the above equation by (1 + i ).
Present value of a growing annuity
In this case each cash flow grows by a factor of (1+g). Similar to the formula for an annuity, the present value of a growing annuity (PVGA) uses the same variables with the addition of g as the rate of growth of the annuity (A is the annuity payment in the first period). This is a calculation that is rarely provided for on financial calculators.
Where i ≠ g :
To get the PV of a growing annuity due, multiply the above equation by (1 + i ).
Where i = g :
Present value of a perpetuity
When
, the PV of a perpetuity (a perpetual annuity) formula becomes simple division.
When this is an increasing perpetuity, this i becomes i’ 1+i’=(1+i)/(1+g) i’=(i-g)/(1+g)
so A/i’ = A x (1+g)/(i-g) not (A/(i-g))
Present value of a growing perpetuity
When the perpetual annuity payment grows at a fixed rate (g) the value is theoretically determined according to the following formula. In practice, there are few securities with precisely these characteristics, and the application of this valuation approach is subject to various qualifications and modifications. Most importantly, it is rare to find a growing perpetual annuity with fixed rates of growth and true perpetual cash flow generation. Despite these qualifications, the general approach may be used in valuations of real estate, equities, and other assets.
This is the well known Gordon Growth model used for stock valuation.
Future value of a present sum
The future value (FV) formula is similar and uses the same variables.
Future value of an annuity
The future value of an annuity (FVA) formula has four variables, each of which can be solved for:
- FV ( A ) is the value of the annuity at time = n
- A is the value of the individual payments in each compounding period
- i is the interest rate that would be compounded for each period of time
- n is the number of payment periods
Future value of a growing annuity
The future value of a growing annuity (FVA) formula has five variables, each of which can be solved for:
Where i ≠ g :
Where i = g :
- FV ( A ) is the value of the annuity at time = n
- A is the value of initial payment at time 0
- i is the interest rate that would be compounded for each period of time
- g is the growing rate that would be compounded for each period of time
- n is the number of payment periods
Derivations
Annuity derivation
The formula for the present value of a regular stream of future payments (an annuity) is derived from a sum of the formula for future value of a single future payment, as below, where C is the payment amount and n the period.
A single payment C at future time m has the following future value at future time n :
Summing over all payments from time 1 to time n, then reversing the order of terms and substituting k = n − m :
Note that this is a geometric series, with the initial value being a = C , the multiplicative factor being 1 + i , with n terms. Applying the formula for geometric series, we get
The present value of the annuity (PVA) is obtained by simply dividing by (1 + i ) n :
Another simple and intuitive way to derive the future value of an annuity is to consider an endowment, whose interest is paid as the annuity, and whose principal remains consta
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